Final answer:
The width of the column 31 inches up is calculated to be 53¼ inches after taking into account the tapering angle of 87 degrees and applying a basic trigonometric relationship.
Step-by-step explanation:
To calculate the width of the column 31 inches up the structure with a taper of 87 degrees, we can use trigonometric relationships to solve the problem. The tapering of the column means that it forms two congruent right-angle triangles, one on each side of the column.
Firstly, the taper angle given is the angle from the base to the column's side, so we should subtract this angle from 90 degrees to obtain the angle at the base of the right-angled triangle, which is 3 degrees (90 - 87).
The width at the base of the column is 55 inches. Since the triangle is right-angled, we can use the tangent function, which is the opposite side (half the difference of the base width and the width 31 inches up) over the adjacent side (31 inches up the column).
For a very small angle like 3 degrees, we can use the approximation that tangent (angle) ≈ angle in radians. There are 57.2958 degrees in a radian, so 3 degrees is approximately 0.05236 radians.
Let the width change be w, then:
- tangent(3 degrees) = w / 31
- 0.05236 ≈ w / 31
- w = 0.05236 * 31
- w ≈ 1.62316 inches
We now subtract this width change from the base width and then divide by 2 to get the width on one side of the column. Then we double that to get the total width.
Total width 31 inches up = (55 - 1.62316) inches
= 53.37684 inches
Finally, we round this figure to the nearest quarter of an inch, which is 53¼ inches.
Therefore, the width of the column at 31 inches up, rounded to the nearest quarter of an inch, is 53¼ inches.